'One of the most ancient arts' is in apposition to 'weaving' (which does not need an article if we are speaking of the activity in general-- like 'no smoking', 'I like skydiving', etc). 'My father, the skydiver, is afraid of heights' -- 'the skydiver' is similarly in apposition to 'my father'. From Tina Blue's webpage: 'An appositive is a substantive that follows another substantive to identify or explain it.
New words, one handy idiom, and a 2-minute quiz — delivered to your inbox to keep your streak alive.